Cauchy Formula Research Articles

On Cauchy Problem Solution for a Harmonic Function in a Simply Connected Domain with Multi-Component Boundary

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

Stages and main tasks of the century-long control theory and system identification development. Part VI. Model predictive control

The article is devoted to one of the relatively new areas in the field of control theory and practice, called model predictive control. The ideas underlying it, the connection with the theory of optimal control and numerical methods for solving optimization problems are described. It should be noted that currently the number of problems solved and control systems developed within the framework of this approach has come out on top in comparison with all others. This is evidenced by numerous publications concerning the theory and practical application of control methods and algorithms using a predictive model. The main distinctive feature of this method is that the control is not synthesized in advance, but is calculated directly during the operation of the system, i.e. at the current time. This made it possible, with such control, to implement original feedback that ensures stabilization of the process when the model is inaccurate and there is uncertainty associated with existing disturbances and measurements. It describes how such feedback is organized based on current measurement data. In this case, there is no need to interpret somehow the nature of the disturbances and errors acting on the system. Much attention in the article is paid to the possibilities that are admitted with this method of control. The model predictive control method seems especially attractive for systems with discrete impulse processes that are well described with using cognitive maps. It is considered an example of cognitive modeling of evolutionary demographic processes in Ukraine and the possibility of control them using a predictive model, when the model contains unstable eigenvalues. When solving problems of predictive control, it is proposed to use trajectory models, which directly follow from the Cauchy formula, in particular for discrete systems, instead of a local description in the state space. This makes it possible to take into account, when solving predictive control problems, how well or poorly controlled, observable or identifiable the system is. Non-standard capabilities of this method with a trajectory description appear during terminal control on a sliding interval with a finite horizon. A number of original formulations of terminal control problems are presented in the article and some of their properties are described.

Distributional Boundary Values of Analytic Functions

Let D be a connected bounded domain in R2, S be its boundary which is closed, connected and smooth. Let Φ(z) = 1 2πi R S ϕ(s)ds s−z , ϕ ∈ X, z = x + iy, X is a Banach space of linear bounded functionals on Hμ, a Banach space of distributions, and Hμ is the Banach space of Hoelder-continuous functions on S with the usual norm. As X one can use also the space Hoelder continuous of bounded linear functionals on the Sobolev space Hℓ on S. Distributional boundary values of Φ(z) on S are studied in detail. The function Φ(t), t ∈ S, is defined in a new way. Necessary and sufficient conditions are given for ϕ ∈ X to be a boundary value of an analytic in D function. The Cauchy formula is generalized to the case when the boundary values of an analytic function in D are tempered distributions. The Sokhotsky-Plemelj formulas are derived for ϕ ∈ X.

Cauchy Formulae and Hardy Spaces in Discrete Octonionic Analysis

In this paper, we continue the development of a fundament of discrete octonionic analysis that is associated to the discrete first order Cauchy–Riemann operator acting on octonions. In particular, we establish a discrete octonionic version of the Borel–Pompeiu formula and of Cauchy’s integral formula. The latter then is exploited to introduce a discrete monogenic octonionic Cauchy transform. This tool in hand allows us to introduce discrete octonionic Hardy spaces for upper and lower half-space together with Plemelj projection formulae.

A note on Cauchy's formula

We use the correlation functions of vertex operators to give a proof of Cauchy's formula∏i=1K∏j=1N(1−xiyj)=∑μ⊆[K×N](−1)|μ|sμ{x}sμ′{y}. As an application of the interpretation, we obtain an expansion of ∏i=1∞(1−qi)i−1 in terms of half plane partitions.

Harmonic and polyanalytic functional calculi on the S-spectrum for unbounded operators

Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy–Fueter operator {mathcal{D}} and of its conjugate overline{{mathcal{D}}}. Thanks to the Fueter extension theorem, when we apply the operator {mathcal{D}} to slice hyperholomorphic functions, we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions, we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the S-spectrum. Another possibility is to apply the conjugate of the Cauchy–Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation, and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the S-spectrum in the quaternionic setting. Fine structures on the S-spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis, and hypercomplex analysis.

Properties of a polyanalytic functional calculus on the S‐spectrum

AbstractThe Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

Super-Schur polynomials for Affine Super Yangian Y( hat{mathfrak{gl}} 1|1)

We explicitly construct cut-and-join operators and their eigenfunctions — the Super-Schur functions — for the case of the affine super-Yangian Y( hat{mathfrak{gl}} 1|1). This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables (pk, θk). The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided.

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S-spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a 0 ( x ) + ∑ j ∈ N y j a j ( x ) , where elements of the parameter vector y = ( y j ) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on U := [ − 1 2 , 1 2 ] . Under the assumption ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ ( D ) < ∞ with some positive sequence ( ρ j ) j ∈ N ∈ ℓ p ( N ) for p ∈ ( 0 , 1 ] we show that for any y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues ( λ j ( y ) ) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap λ 2 ( y ) − λ 1 ( y ) is uniformly positive in U ∞ . From this, we prove the holomorphic extension property of λ 1 ( y ) to a complex domain in C ∞ and estimate partial derivatives of λ 1 ( y ) with respect to the parameter y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ 1 ( y ) .

On a problem for a delay differential equation

The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem.

Optical, electronic, and structural properties of ScAlMgO4

Magnesium aluminate scandium oxide (ScAlMgO4) is a promising lattice-matched substrate material for GaN- and ZnO-based optoelectronic devices. Yet, despite its clear advantages over substrates commonly used in heteroepitaxial growth, several fundamental properties of ScAlMgO4 remain unsettled. Here, we provide a comprehensive picture of its optical, electronic and structural properties by studying ScAlMgO4 single crystals grown by the Czochralski method. We use variable angle spectroscopic ellipsometry to determine complex in-plane and out-of-plane refractive indices in the range from 193 to 1690 nm. An oscillator-based model provides a phenomenological description of the ellipsometric spectra with excellent agreement over the entire range of wavelengths. For convenience, we supply the reader also with Cauchy formulas describing the real part of the anisotropic refractive index for wavelengths above 400 nm. Ab initio many-body perturbation theory modeling provides information about the electronic structure of ScAlMgO4, and successfully validated experimentally obtained refractive index values. Simulations also show exciton binding energy as large as a few hundred of meV, indicating ScAlMgO4 as a promising material for implementation in low-threshold, deep-UV lasing devices operating at room temperature. X-ray diffraction measurements confirm lattice constants of ScAlMgO4 previously reported, but in addition, reveal that dominant crystallographic planes (001) are mutually inclined by about 0.009{\deg}. In view of our work, ScAlMgO4 is a highly transparent, low refractive index, birefringent material similar to a sapphire, but with a much more favorable lattice constant and simpler processing.

The mathcal {F}-Resolvent Equation and Riesz Projectors for the mathcal {F}-Functional Calculus

The Fueter-Sce-Qian mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to slice hyperholomorphic functions. By means of the Cauchy formula for slice hyperholomorphic functions it is possible to have a Fueter-Sce-Qian mapping theorem in integral form for n odd. On this theorem it is based the mathcal {F}-functional calculus for n-tuples of commuting operators. It is a functional calculus based on the commutative version of the S spectrum. Furthermore, it is a monogenic functional calculus in the spirit of McIntosh and collaborators. In this paper, inspired by the quaternionic case and some particular Clifford algebras cases, we show a general resolvent equation for the mathcal {F}-functional calculus in the Clifford algebra setting. Moreover, we prove that the mathcal {F}-resolvent equation is the suitable equation to study the Riesz projectors.

Of shadows and gaps in spatial search

Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty \etal ({\em Physical Review A}, {\bf 102}:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

Towards a general [formula omitted]-resolvent equation and Riesz projectors

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to write the Fueter-Sce-Qian extension theorem in integral form and to define the F-functional calculus for n-tuples of commuting operators. This functional calculus is defined on the S-spectrum and generates a monogenic functional calculus in the spirit of McIntosh and collaborators. One of the main goals of this paper is to show that the F-functional calculus generates the Riesz projectors. The existence of such projectors is obtained via the F-resolvent equation which was previously known only in the quaternionic setting and also its existence was under question. In this paper we prove the F-resolvent equation in the Clifford algebra setting. It is much more complicated than the one in the quaternionic case since it contains various pieces, however it still allows to nicely define the Riesz projectors.

Linear/nonlinear optical properties and dispersion parameters of nanocrystalline indigo organic semiconductor films

Films of indigo dye as organic semiconductors have been prepared with different thicknesses (462–885 nm) through thermal evaporation. X-ray diffraction shows a nanocrystalline structure for the films with crystallite size 10.5 ± 0.1 nm. AFM confirms the nanocrystalline structure of the prepared films. The optical properties of the indigo films have been investigated using Swanepoel's method. These films have 2 optical energy gaps (1.67 and 2.61 eV) and Urbach's energy of 0.22 eV. The refractive index has normal behavior to decrease with increasing the wavelength following the Cauchy formula and the single oscillator model. The oscillator energy E o and dispersion energy E d are 2.75 and 4.81 eV, respectively. The lattice dielectric constant ε L is 2.89 while the static dielectric constant ε s is 2.75. The linear/nonlinear optical susceptibility and nonlinear refractive index have similar behavior to increase with increasing the photon energy. • Thermal evaporated films of indigo have a nanocrystalline structure. • Indigo films have two indirect optical transitions. • The optical properties of the indigo films are thickness independent. • The nonlinear refractive index and linear/nonlinear optical susceptibility increase with the photon energy.

New trends in control theory

The article outlines the conceptual foundations of new trends in control theorythat have been intensively developing in recent years. Unlike classical controltheory, which was formed in the last century and is based on well-known mathematical models of controlled processes in the form of local equations, new approaches to linear stationary systems use input-output relations that follow directly from the Cauchy formula for both continuous and discrete systems. On thebasis of the same description, it is possible to substantiate and obtain the socalled data-based models, which are directly linked to data that form, at the observation intervals, the trajectories of already implemented past processes andfuture ones, for which control is to be synthesized. This approach is focusedprimarily on finding control from the prediction model. At the same time, thecurrent measurements carried out at the plant make it possible to implementfeedback and, in case of discrepancies between the forecast and the real process,to correct the predictive control, i.e. such a way to stabilize it. Control by trajectory prediction model allows to exclude model identification by trajectory data,and control directly on their base. Since the data contain errors, the most important issue in the considered approach is the robustness of the chosen control.A large number of published works are dedicated to this problem, where theguaranteed approach, focused on the worst-case in the data, is the most in demand. In most cases, control synthesis is reduced to solving various optimizationproblems, mainly on the finite prediction horizon. Considerable attention in thearticle is paid to methods for solving synthesis problems based on SVD decomposition. To reduce the complexity of the tasks to be solved, it is proposed to reduce it to terminal control on the horizon of a short duration. Then an iterativecontrol strategy is implemented, which, due to feedback, ensures the feasibilityof the global control goal.

Triple-wavelength quantitative phase imaging with refractive index measurement

A method to quantitatively measure the refractive index (RI) and the topography of transparent samples is proposed. Three quantitative phase images at different wavelengths are firstly obtained by using a quadriwave lateral shearing interferometry (QLSI) technique, and then the RIs at the three wavelengths and the physical thickness distribution of the sample are independently calculated with the help of Cauchy's dispersion formula. Neither the highly dispersive medium nor the manual operation of changing the surrounding medium is required. From the measured RIs, the composition of the sample can be identified besides the topography of the sample. Both simulation and experimental results verified the effectiveness and feasibility of the proposed method.

The Poisson kernel and the Fourier transform of the slice monogenic Cauchy kernels

The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure to extend holomorphic functions of one complex variable to slice monogenic functions and to monogenic functions. Using the Cauchy formula of slice monogenic functions the FSQ-theorem admits an integral representation for n odd. In this paper we show that the relation Δn+1(n−1)/2SL−1=FnL between the slice monogenic Cauchy kernel SL−1 and the F-kernel FnL, that appear in the integral form of the FSQ-theorem for n odd, holds also in the case we consider the fractional powers of the Laplace operator Δn+1 in dimension n+1, i.e., for n even. Moreover, this relation is proven computing explicitly the Fourier transform of the kernels SL−1 and FnL as functions of the Poisson kernel. Similar results hold for the right kernels SR−1 and of FnR.

Cauchy’s surface area formula in the Heisenberg groups

We show an analogue of Cauchy's surface area formula for the Heisenberg groups $\mathbb{H}\_n$ for $n\geq 1$, which states that the p-area of any compact hypersurface $\Sigma$ in $\mathbb{H}\_n$ with its p-normal vector defined almost everywhere on $\Sigma$ is the average of its projected p-areas onto the orthogonal complements of all p-normal vectors of the Pansu spheres (up to a constant). The formula provides a geometric interpretation of the p-areas defined by Cheng–Hwang–Malchiodi–Yang in $\mathbb{H}\_1$ and Cheng–Hwang– Yang in $\mathbb{H}\_n$ for $n\geq 2$. We also characterize the projected areas for rotationally symmetric domains in $\mathbb{H}\_n$; namely, for any rotationally symmetric domain with boundary in $\mathbb{H}\_n$, its projected p-area onto the orthogonal complement of any normal vector of the Pansu spheres is a constant, independent of the choice of the projected directions.